The Research On The Kirchhoff Index Problem Of Some Chemical Graphs

The spectral theory is an important research field of combinatorial matrix theory and graph theory.And in many cases,the parameters based on the normalized Laplacian can reflect the structure and properties of the graph more truly than other matrices,so many researchers pay more and more attention to the normalized Laplacian of the graph.At the same time,due to its close relationship with spectral geometry and random walk,the study of canonical Laplacian matrix of graphs has gradually become a hot topic of spectral theory of graphs in recent years.Consider the linear[n]phenylenes Ln6,4,consisting of n six-membered and n four-membered rings.Then the Mobius phenylenes chain Hn(6,4)is the graph obtained from the Ln6,4 by identifying the opposite lateral edges in reversed way,whereas the cylinder phenylene chain H'n(6,4)is the graph obtained from the Ln6,4 by identifying the opposite lateral edges in ordered way.Based on the normalized Laplacian matrix decomposition theorem,we obtain degree-Kirchhoff index and spanning trees expression of the Mobius type hexagon chain Hn(6,4),and get Kirchhoff index and spanning tree formula of the cylinder phenylene chain H'n(6,4).The main contents of this paper include:In Chapter 1,we briefly introduces the research background and current situation of this paper.In Chapter 2,the calculation formulas of the degree-kirchhoff index and spanning trees of the Mobius type hexagon chain are derived.And found that the degree-kirchhoff is Gutman index 1/3 times.In Chapter 3,we give the Kirchhoff indices and spanning trees of Mobius phenylenes chain and cylinder phenylenes chain.In Chapter 4,we give conclusions.Figure[2]table[4]reference[55]...